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Square
A square is an equilateral (equally-lengthed sides) and equiangular quadrilateral. A square is both a rhombus and a rectangle, simultaneously. Thus, a square shares the properties of each: * Quadrilateral polygon * All four sides are of equal length (congruent) * All four corner angles are of equal measure (congruent) ** All four corner angles are right angles ** Adjacent sides meet at right angles ** Opposite sides are parallel to one another *** Parallelogram * Diagonals bisect the angles of the corners they connect at one-half a right angle * Diagonals bisect one another * Diagonals are of equal length * Diagonals intersect at right angles In Euclidean geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles (90 degree angles, or right angles). A square with vertices ABCD would be denoted . Classification Two-dimensional object made up with four points, and four equal line-segments. The mensuration formula The perimeter of a square whose sides have length t'' is : P=4t. And the area is : A=t^2. In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term ''square to mean raising to the second power. Standard coordinates The coordinates for the vertices of a square centered at the origin and with side length 2 are (±1, ±1), while the interior of the same consists of all points (x''0, ''x''1) with −1 < ''xi'' < 1. Equations The equation max (x^2, y^2) = 1 describes a square. This means " x^2 or y^2 , whichever is larger, equals 1." The circumradius of this square is \sqrt{2} . Properties The diagonals of a square bisect each other. The diagonals of a square bisect its angles. The diagonals of a square are perpendicular. Opposite sides of a square are both parallel and equal. All four angles of a square are equal. ''(Each is \frac{360}{4} = 90 degrees, so every angle of a square is a right angle.) The diagonals of a square are equal. Other facts *If the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are \sqrt{2} (about 1.414) times the length of a side of the square. This value, known as Pythagoras’ constant, was the first number proven to be irrational. *A square can also be defined as a rectangle with all sides equal, or a rhombus with all angles equal, or a parallelogram with equal diagonals that bisect the angles. *If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square. (Rectangle (four equal angles) + Rhombus (four equal sides) = Square) *If a circle is circumscribed around a square, the area of the circle is \pi/2 (about 1.57) times the area of the square. *If a circle is inscribed in the square, the area of the circle is \pi/4 (about 0.79) times the area of the square. *A square has a larger area than any other quadrilateral with the same perimeter (http://www2.mat.dtu.dk/people/V.L.Hansen/square.html). *A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon). *The square is in two families of polytopes in two dimensions: hypercube and the cross polytope. The Schläfli symbol for the square is {4}. *The square is a highly symmetric object (in Goldman geometry). There are four lines of reflectional symmetry and it has rotational symmetry through 90°, 180° and 270°. Its symmetry group is the dihedral group D_4 . Non-Euclidean geometry In non-euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles. In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger squares have smaller angles. Examples: See also *Cube (3-dimensional square) *Pythagorean theorem *Square lattice *Unit square External links * Square Calculation * Animated course (Construction, Circumference, Area) * *Definition and properties of a square With interactive applet *Animated applet illustrating the area of a square af:Vierkant ar:مربع an:Cuadrato arz:مربع ast:Cuadráu ay:Pusi k'uchuni az:Kvadrat bn:বর্গক্ষেত্র be:Квадрат be-x-old:Квадрат bs:Kvadrat bg:Квадрат ca:Quadrat (polígon) cs:Čtverec cy:Sgwâr da:Kvadrat de:Quadrat (Geometrie) et:Ruut el:Τετράγωνο es:Cuadrado eo:Kvadrato (geometrio) eu:Lauki fa:مربع fr:Carré gl:Cadrado ko:정사각형 hi:वर्गाकार hr:Kvadrat io:Quadrato id:Persegi is:Ferningur it:Quadrato (geometria) he:ריבוע ka:კვადრატი sw:Mraba ht:Kare lo:ຮູບຈັດຕຸລັດ la:Quadrum lv:Kvadrāts lt:Kvadratas li:Veerkant hu:Négyzet mk:Квадрат ml:സമചതുരം mr:चौरस mn:Квадрат nl:Vierkant (meetkunde) ja:正方形 no:Kvadrat nn:Kvadrat uz:Kvadrat km:ការ៉េ pl:Kwadrat pt:Quadrado ro:Pătrat qu:T'asra ru:Квадрат sco:Squerr scn:Quatratu simple:Square (geometry) sk:Štvorec sl:Kvadrat (geometrija) szl:Kwadrat sr:Квадрат sh:Kvadrat su:Pasagi bener fi:Neliö (geometria) sv:Kvadrat tl:Parisukat ta:சதுரம் th:รูปสี่เหลี่ยมจัตุรัส vi:Hình vuông tr:Kare uk:Квадрат ur:مربع (ہندسہ) vls:Vierkant yi:קוואדראט zh-yue:正方形 bat-smg:Kvadrots zh:正方形//// Category:Geometric shapes Category:Quadrilaterals Category:Polygons Category:Geometry